#7149002

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993875
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#60836	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1798 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(3,7)(6,8)(9,10,15,19,14,21,13,20)(11,22,30,33,26,32,25,16)(12,18,27,17,29,36,28,34)(23,38,39,35,40,31,24,37);(2,5)(3,7)(9,11,15,30,14,26,13,25)(10,16,20,32,21,33,19,22)(12,23,28,24,29,40,27,39)(17,31,36,37,34,38,18,35);(1,2)(3,6)(4,5)(7,8)(9,12)(10,17)(11,23)(13,27)(14,29)(15,28)(16,31)(18,19)(20,36)(21,34)(22,35)(24,30)(25,39)(26,40)(32,37)(33,38);(2,5)(6,8)(9,12)(10,18)(11,24)(13,28)(14,29)(15,27)(16,31)(17,19)(20,34)(21,36)(22,37)(23,30)(25,40)(26,39)(32,35)(33,38);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=gi,b^2=fi,e^2=h,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7056198);